From a group of 2 men and 3 women , two persons are selected . Describe the sample space of the experiment , if E is the event in which one man and one women are selected , then which are the cases favourable to E?

Answer

To solve the problem step by step, we will first identify the sample space for the selection of two persons from a group of 2 men and 3 women. Then, we will identify the favorable cases for the event E, where one man and one woman are selected.

### Step 1: Identify the Group
We have:
- Men: M1, M2
- Women: W1, W2, W3

### Step 2: Determine the Sample Space
The sample space (S) consists of all possible combinations of selecting 2 persons from the group. We can select either:
1. Two men
2. Two women
3. One man and one woman

Let's list all possible combinations:
1. Selecting both men: (M1, M2)
2. Selecting two women: (W1, W2), (W1, W3), (W2, W3)
3. Selecting one man and one woman:
- (M1, W1)
- (M1, W2)
- (M1, W3)
- (M2, W1)
- (M2, W2)
- (M2, W3)

Thus, the complete sample space S is:
\[ S = \{ (M1, M2), (W1, W2), (W1, W3), (W2, W3), (M1, W1), (M1, W2), (M1, W3), (M2, W1), (M2, W2), (M2, W3) \} \]

### Step 3: Identify the Event E
Event E is defined as the event of selecting one man and one woman. From our sample space, we can identify the combinations that satisfy this condition:
- (M1, W1)
- (M1, W2)
- (M1, W3)
- (M2, W1)
- (M2, W2)
- (M2, W3)

### Step 4: List the Favorable Cases
The favorable cases for event E are:
\[ E = \{ (M1, W1), (M1, W2), (M1, W3), (M2, W1), (M2, W2), (M2, W3) \} \]

### Final Answer
The sample space S is:
\[ S = \{ (M1, M2), (W1, W2), (W1, W3), (W2, W3), (M1, W1), (M1, W2), (M1, W3), (M2, W1), (M2, W2), (M2, W3) \} \]

The favorable cases for event E (one man and one woman) are:
\[ E = \{ (M1, W1), (M1, W2), (M1, W3), (M2, W1), (M2, W2), (M2, W3) \} \]